TR10-122 Authors: Zhixiang Chen, Bin Fu, Yang Liu, Robert Schweller

Publication: 2nd August 2010 05:38

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This paper is our second step towards developing a theory of

testing monomials in multivariate polynomials. The central

question is to ask whether a polynomial represented by an

arithmetic circuit has some types of monomials in its sum-product

expansion. The complexity aspects of this problem and its variants

have been investigated in our first paper by Chen and Fu (2010),

laying a foundation for further study. In this paper, we present

two pairs of algorithms. First, we prove that there is a

randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in

an $n$-variate polynomial of degree $k$ represented by an

arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time

algorithm is devised when the circuit is a formula, here $p$ is a

given prime number. Second, we present a deterministic $O^*(2^k)$

time algorithm for testing multilinear monomials in

$\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a

randomized $O^*(1.5^k)$ algorithm is given for these polynomials.

The first algorithm extends the recent work by Koutis (2008) and

Williams (2009) on testing multilinear monomials. Group algebra

is exploited in the algorithm designs, in corporation with the

randomized polynomial identity testing over a finite field by

Agrawal and Biswas (2003), the deterministic noncommunicative

polynomial identity testing by Raz and Shpilka (2005) and the

perfect hashing functions by Chen {\em at el.} (2007). Finally, we

prove that testing some special types of multilinear monomial is

W[1]-hard, giving evidence that testing for specific monomials is

not fixed-parameter tractable.